1.6 Prime and Composite Numbers

September 3, 20131.6 Prime and Composite NumbersMath Message 1.6Please complete the following Math Message in your Math notebooks and have your answers ready to share:

Draw ALL possible rectangular arrays for these numbers: 2, 4, 5, 10, 11, and 16What exactly are prime numbers?A prime number has exactly two factors- 1 and the number itselfImportant to know: the number 1 is considered neither prime nor composite

Lets look at some examples.

2 1 and 2 17 17 and 1 23 1 and 23

What are composite numbers?A composite number has more than two factors

Some examples of composite numbers:

Number Factors 4 1, 2, 4 16 1, 2, 4, 8, 16 35 1, 5, 7, 35September 4, 2013 1.7 Square NumbersMath Message 1.7Use coins, M & M candy, or any other items that resemble counters. Try and make a rectangular array with an equal number of rows and columns for each of the following numbers:141618

Which numbers make this kind of array? Show your work for each array. Answer to the Math Message:Of the three numbers, 16 is the only one that can be represented equally

4 x 4 array * * * * * * * * * * * * * * * *How is a 4 x 4 array similar to a square?When an array has the same number of rows and columns, it is shaped like a square and is called a square arrayThe number it represents is called a square number

Since 16 can be represented by a square array, the number 16 is a square numberFinding other Square NumbersHow many objects are in the array? 9How many rows are in the array set? 3How many columns are in the array set? 3

3-by-3 array Square NumbersAny square number can be written as the product of a number multiplied by itself1 x 1 = 12 x 2 = 43 x 3 = 94 x 4 = 16

A shorthand way of writing square numbers looks like this: 9 = 3 x 3 = 32

Lets work on MJ pg. 20

What are exponents? **You can read 32 as 3 times 3, 3 squared, or 3 to the second powerThe raised 2 is called an exponent. It tells you that 3 is used as a factor 2 times 32 = 3 x 3 = 9Exponential Notation- numbers that are written with an exponent ex. 32

What is the difference between doubling a number and squaring a number?What do you get when you double 10? 20 = 10 +10 or 10 x 2

What do you get when you square 10? 10 * 10 = 100 or 102

September 5, 20131.8 Unsquaring Numbers1.8 Math MessageFind the numbers that make these statements true. Write out and complete each statement in your Math notebook.

1) ______ * ______ = 4

2) ______ 2 = 81

What numbers could the blank spaces in these problems represent?Answers to the Math Message:

The factors of 1 and 4, or 2The factor 9

Now, lets replace the blank spaces with the letter (the variable), N. What number is being represented by the variable, N?

N * N = 4 N = 2What do you know about variables?A variable can only represent one number if the number sentence is true.

For example: m2 = 81 m squared equals 81 m * m = 81

What number is being represented by the variable m to make the number sentence true?

m = 9 Unsquaring NumbersWhen unsquaring a number, we need to undo the operation

If you square a number, you are multiplying it by itself to get the product 4 * 4 = p

When you are given the product, you will need to undo the multiplication to identify the number that was squared n * n = 16 Problems well complete in class..What number, multiplied by itself, is equal to 289?

What strategies are you using?

Lets practice unsquaring numbers 19610,0007,225441Finding the Square Root of NumbersWhen you unsquare a number, you have found the square root of the number

What number squared is 64? 8

So, what is 64 squared? 8, because 8 * 8 = 64

What is the square root of 64? 8Testing our results using a calculatorWhen using a calculator to find the square root you will need to know the following:1) If the display shows a whole number, then the original number is a square number. *For example: 576 is a square number because using the square-root key displays a whole number, 242) If the display shows a decimal, then the original number is not a square number. * For example: 794 is not a square number because using the square-root key displays a decimal--- 28.178006 (rounded to 6 decimal places) September 6, 20131.9 Factor Strings and Prime Factorizations1.9 Math Message8 + 8 and 4 * 4 are two names for the number 16. In your Math notebook, write at least five other names for 16. Brainstorm the different ways and be prepared to share your answers

Name-Collection Box for 16 What are Factor Strings?A factor string is a multiplication expression that has at least two factors that are greater than 1. *In a factor string, the number 1 may NOT be used as a factor

Example of a factor string for the number, 24: 24 is 2 * 3 * 4 In the factor string, you see three factors (2, 3, 4), this is called, the length of the factor string *Now, lets find other factor strings for 24Lets practice(in class)Find the factor string for 7What type of number is 7? A prime numberThe number 1 may not be used in a factor string, so there are no factor strings for prime numbers3) Lets find all possible factor strings for 36Is 2 a factor of 36? Yes, 36 = 2 * 18Is 2 a factor of 18? Yes, 36 = 2 * 2 * 9Is 2 a factor of 9? NoIs 3 a factor of 9? Yes, 36 = 2 * 2 * 3 * 3 Prime FactorizationWhat kind of numbers are the factors in the longest possible factor string for any number? Prime numbersThe longest factor string for a number is called the prime factorization of the number For example: The prime factorization of 24 is 2 * 2 * 2 * 3Using Factor Trees to find all the prime factorsWe will practice creating a Factor Tree for 36

36 6 * 6 3 * 2 3 * 2

36 = 2 * 2 * 3 * 3

36 = 22 * 32